Range of researches in theoretical physics

Interests and results are presented

Dr. A. Mazurenko

Obtaining of the exact analytical solutions for steady-state Schrodinger equations ( C₀ є - h² /(8 p² m) is constant)

ж
з
и

C₀

d²

d x²

+ U(x)

ц
ч
ш

Y(x) = E Y(x)

(1)

on a plane

ж
з
и

C₀

d²

d x²

+ C₀

d²

d y²

+ U(x,y)

ц
ч
ш

Y(x,y) = E Y(x,y)

(2)

in space

( C₀ D + U(r)) Y(r) = E Y(r)

(3)

for new special cases of potential functions U(r) . The function U(r) depends on both module and direction of the multidimensional vector r .

To solve the equations the author specially constructed functions f_n (z) which are orthogonal on a contour L

у
х
L

f_n (z) f_m (z) dM(z) = d_m, n d_n²

(4)

with a measure M(z) .

Functions generalizing orthogonal polynomials of the Jacobi P_n^(a, b) (x) of continuous argument x are determined.

The polynomials p_n (x,b) ( f_n (z) є p_n (z,b) ) which are orthogonal on interval [-1, 1] (the contour L is non-closed and coincides with interval [-1, 1] ) with respect to different weight functions s(x,b) ( M(z) = т₀^z s(x,b) dx + C ) have been constructed.

Name of constructed polynomials p_n (x,b) Weight s(x,b) in orthogonality relation
Christoffel-Legendre-1

s(x,b)   =    (b-x)
b
  ,    b і 1

Christoffel-Jacobi-1

s(x,b)   =    (b-x)
b
  (1-x)^a(1+x)^b,    b і 1,    a > -1,    b > -1

Uvarov-Legendre-1

s(x,b)   =    b
(b-x)
  ,    b і 1

Christoffel-Legendre-2

s(x,b)   =    (b²-x²)
b²
  ,    b і 1

Uvarov-Legendre-2

s(x,b)   =    b²
(b²-x²)
  ,   b і 1

Christoffel-Legendre-4

s(x,b)   =    (b₁² -x²) (b₂² -x² )
(b₁ b₂ )²
  ,    м
н
о

b₁ і 1 and b₂ і 1

"b О Re     b = b₁ є b₂

Christoffel-Krawtchouk and nonclassical Gegenbauer polynomials f_n (z = m) є p_n (m) with discrete weights s(m) have been constructed. The orthogonality relation for polynomials of a discrete variable has a stepwise measure

M(z) =

z
у
х
-Ґ

е
< m

s(m) d(x-m) dx + C є

е
< m

s(m) q(x-m) + C

(5)

where d(z) є [d/d z] q(z) is the Dirac delta-function and q(z) is the Heaviside function

q(z) є

м
н
о

1 , if z і 0 ;

0 , if z < 0 .

(6)

Latex source file: theorph.zip 1,5 kb.

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